The number of values of a for which the curves 4 x 2 + a 2 y 2 = 4 a 2 and y 2 = 16 x are orthogonal is

Step-by-step solution

Given curves are 4 x 2 + a 2 y 2 = 4 a 2 ...(i) and y 2 = 16 x ...(ii) If the curves intersect at P α , β , then α 2 a 2 + β 2 4 = 1 and β 2 = 16 α . On differentiating equation (i), we get, 2 x a 2 + 2 y 4 y ' = 0 ⇒ y ′ = − 4 x a 2 y ⇒ m 1 = - 4 α a 2 β On differentiating equation (ii), we get, 2 y y ' = 16 ⇒ m 2 = 8 β For curves to be orthogonal, m 1 m 2 = - 1 i.e. - 4 α a 2 β 8 β = - 1 ⇒ 32 α = a 2 β 2 &#865